Models of $VT{C}^0$ as exponential integer parts

We prove that (additive) ordered group reducts of nonstandard models of the bounded arithmetical theory $\mathsf{V}\mathsf{T}{\mathsf{C}}^{\mathsf{0}}$ are recursively saturated in a rich language with predicates expressing the integers, rationals, and logarithmically bounded numbers. Combined with our previous results on the construction of the real exponential function on completions of models of $\mathsf{V}\mathsf{T}{\mathsf{C}}^{\mathsf{0}}$, we show that every countable model of $\mathsf{V}\mathsf{T}{\mathsf{C}}^{\mathsf{0}}$ is an exponential integer part of a real-closed exponential field.